The instructor wears a mike (connected to a little box worn at belt level) transmitting to an amplifying system and through it to the speakers. During the demos the camera gets the audio through the air from the speakers (I hope my English is not too terrible here :) and everything is OK but when the instructor moves around he turns off the mike, people gather around him and I can't get clear audio. This is a pity because we miss the best parts, the corrections people could really learn from.

The suggested solution was to get all of the audio from the PA mixer, and when the instructor does his walkabout, to mute the signal to the speakers instead of muting the mic.

This, however, has two problems: First, a person would have to man the mixer, cutting off the sound to the speakers whenever the speakers should be shut off. Second, as the camera is moving around and is not stationary at the mixer, we would have to record the audio at the mixer and later sync it to the video - quite a lot of work.

So my question is: Is there a way to wirelessly route the audio directly from a microphone worn by the instructor to my camera's microphone input, without having to route it to the mixer, record it there and sync it with the video at later processing?

No matter how I think about it someone will have to be at the mixing desk momentarily.

If syncing the audio in post isn't an option, then I'd get another set of wireless transmitters, hook the transmitter to the mixing desk and connect the receiver to your camera. Then when the instructor does the demonstrations you could quickly go to the mixing desk and mute the output to the speakers.

This way you would also be getting much clearer audio because you are recording the instructors mic signal the entire time.

That being said, syncing audio in post isnt as hard or as much of a hassle as it use to be. You can use software like Plural Eyes to easily sync recorded audio to reference audio in the camera. I use this software all the time and it has never failed.